Why don't airlines install Faraday cages in their planes?

Airplane safety addresses the proverbial "one in a billion risk" (which is not an exaggeration as I explain below) that a problem can occur, it doesn't deal with the normal situations where typically all will be fine. So, the fact that you won't expect problems with cell phone use in planes as pointed out in Moo's answer is not all that relevant. What matters when considering safety is if theoretically there exists a loophole that could lead to problems. One then has to make sure that such loophole cannot actually occur due to further detailed studies or by directly exposing a test flight to the theoretical risk and proving that it doesn't pose a problem.

The loophole that makes it at the moment unsafe to let people use cell phones involves by many passengers in a plane using their phones at the same time when they all have bad reception. It is precisely the shielding due to the plane's airframe that causes poor reception. So, suppose everyone were to switch on their phones before landing (say the flight was delayed and everyone is calling home to tell that they'll arrive in half an hour). Then because the phones have very poor reception they'll crank up their output power to 1 to 2 Watt. You'll then have hundreds of phones transmitting a few hundred Watts in total inside the airframe. The electromagnetic field strength in the plane can then build up large values, each point in the plane will be exposed via direct line of sight transmissions or indirectly via reflections.

Each phone is itself blasted by the RF transmissions of all the other phones, which then causes intermodulation effects. The signal form other phones will enter your phone and that signal will be added to the signal produced by your own phone before the final amplification stage. This leads to a new signal at a frequency given by the sum or differences of the signals that are amplified. The most problematic of these are third order intermodulation signals that are the sum of two mines one other, as the frequency of that signal will be of the same order as the intended signals and these are then not blocked by the device itself.

So, the net effect is that all the phones will end up transmitting not just on their normal channels, but they'll also output strong signals at other frequencies, and these signals can then interfere with the aircraft's communication signals directly, or indirectly by causing intermodulation effects in the pilot's receiver. While there are many filters in receivers and transmitters that block unwanted signals, they have their limits, particularly for rejecting unwanted signals close to the frequency of the desired signal.

But suppose that the c**kpit has a good RF shielding, would that not prevent problems? That's unlikely to work because of the many electric cables moving from the c**kpit through the cabin. These will carry the electromagnetic fields inside the c**kpit. We need to keep in mind that the plane's antenna is capturing a signal of the order of a microwatt of power and then the hundreds of watts of interference (that's not in the form of a clean signal at a totally different frequency) is going to pose a huge problem, even for the best electronics available today.

While this objection may seem far fetched, it cannot be ruled out at present that in exceptional cases you get an intermodulation signal at an ATC frequency or at a navigational beacon frequency causing a catastrophic outcome. We don't want an accident caused by cell phone induced RF interference effects every few years, but with more than 100,000 take-offs and landings per day, more than 100 million in a 3 year period, you're aiming to rule out one a billion freak incidents.

Aviation electronics already are shielded to rigorous standards, such as RTCA DO-160 - the issue with cell phones and other transmitting devices is largely based on out dated scientific understanding and studies, and older requirements.

In the US, both the FCC (which is the body which actually bans cell phone use on aircraft due to issues with ground cell towers rather than interference with avionics - civil aircraft move quick enough that hand offs between cell towers can become too frequent and problematic) and the FAA (which leaves cell phone use up to airlines through exceptions to FAR 14 C.F.R § 91.21) have indicated they are willing to relax rules and change official stances, even on older aircraft.

The issue these days with cell phones on civil aircraft has rapidly shifted to social issues rather than technical issues - people don't want to be sat next to the "cell phone thug" who insists on shouting into his device for the entirety of a five hour flight.